Modern family: Addition

State level Olympiad (2)

Q 5 :  If a,b,c,d,e are positive numbers bounded by p and q, i.e, if they lie in [p,q], 0 < p,
prove that (a+b +c +d +e) ({1}/{a} + {1}/{b} + {1}/{c} + {1}/{d} +
{1}/{e}) <= 25 + 6 ( √{ {p}/{q}} -
√ { {q}/{p}})^2 and determine when there is equality.

Solution 5 : Fix four of the variables and allow the other to vary. Suppose, for example, we fix all but x. Then the expression on the LHS has the form (r + x)(s + {1}/{x}) = (rs + 1) + sx + {r}/{x}, where r and s are fixed.
But this is convex. That is to say, as x increases if first decreases, then increases. So its maximum must occur at x = p or x = q. This is true for each variable.

Suppose all five are p or all five are q, then the LHS is 25, so the inequality is true and strict unless p = q. If four are p and one is q, then the LHS is 17 + 4 ( {p}/{q} + \{q}/{p} ).

Similarly if four are q and one is p. If three are p and two are q, then the LHS is 13 + 6 ( {p}/{q} + \{q}/{p} ). Similarly if three are q and two are p.

( {p}/{q} + {q}/{p} ) >= 2 with equality iff p = q, so if p < q, then three of one and two of the other gives a larger LHS than four of one and one of the other. Finally, we note that the RHS is in fact 13 + 6 ( {p}/{q} + {q}/{p} ), so the inequality is true with equality iff either (1) p = q or (2) three of v, w, x, y, z are p and two are q or vice versa.

Q 6 :  If A and B are fixed points on a given circle and XY is a variable diameter of the same circle, determine the locus of the point of intersection of lines AX and BY. You may assume that AB is not a diameter.

Solution 6 : Assume that the circle is the unit circle centered at the origin. Then the points A and B have coordinates (- a,b) and (a,b) respectively and X and Y have coordinates (r,s) and (- r,- s). Note that these coordinates satisfy a^2 + b^2 = 1 and r^2 + s^2 = 1 since these points are on a unit circle. Now we can find equations for the lines:
AX -------> y  =  {(s- b)x+rb+sa}/{r+a}

BY ---------> y = {(s+b)x+rb- sa}/{r+a}.

Solving these simultaneous equations gives coordinates for P in terms of a, b, r, and
s: P = ( {as}/{b},{1 - ar}/{b} ). These coordinates can be parametrized in Cartesian variables as follows:

x = {as}/{b}
y = {1 - ar}/{b}.

Now solve for r and s to get r = {1-by}/{a} and s = {bx}/{a} . Then since r^2 + s^2 = 1, ( {bx}/{a} )^2 + ( {1-by}/{a} )^2 = 1; which reduces to x^2 + (y- 1/b)^2 = {a^2}/{b^2}.

This equation defines a circle and is the locus of all intersection points P. In order to define this locus more generally, find the slope of this circle function using implicit differentiation:

2x + 2(y- 1/b)y' = 0
(y- 1/b)y' = - x 
y' &= {- x}/{y- 1/b}.

Now note that at points A and B, this slope expression reduces to
y' = {- b}/{a} and y' = {b}/{a} respectively, values which are identical to the slopes of lines AO and BO. Thus we conclude that the complete locus of intersection points is the circle tangent to lines AO and BO at points A and B respectively.

---------

I was able to finish whole paper of 2 hours 20 mins in 1.5 hours and after that I sat there to recheck my whole paper to see if any mistakes have been committed or steps have been omitted. After double check reassurance; I submitted my answer sheet followed by exited the premises.

Outside dad was waiting for me.

Phil : So, buddy, how was exam?

Me : It was good. Easy for me. Let's rush straight to hotel I'm quite hungry.

Phil : Sweet.

After that we arrived at our hotel. I ate my fill and started my revision for Physics exam. Time passed, I stayed in my room but Dad did travel on his own accord.
Hotel services were quite good. Next exam was 2 days after first one. And with my perfect recall and photographic memory, although it wasn't necessary but I still revised my subject materials and solved problems from my books.

Meanwhile some other thoughts were also buzzing in my head. Eg. Star wars (Episode iv) is going to be entering into final lap (In this life/world First star wars movie is little delayed on schedule compared to original), Fed is going to increase interest rates (20%) soon, Oil crisis is also nearing, Recession (1980-81) is also nearing, Latin American debt crisis etc. etc.

To extract significant profits from these defaults I also have to gear up my preparation to enter into final stage of launch readiness. In all of these aspects main fuel is going to be cash flow and in order for me to not face cash crunch; expansion of DWD financial services (bank owned by me) is of utmost importance.

And since it is easy to buy an infrastructure than to build one; in order to reduce necessary work pertaining to lack of most importantly time and then efforts; my brain is beginning to a sketch a plan for successful takeover/acquisition of another bank in order to satiate our needs.

For that to happen, a suitable target must be selected, necessary conditions must be present and if not then must be created in order for successful takeover. And since it's not my style to settle for lesser if I can achieve better; I decided to learn for the expansion/acquisition history of JP Morgan Chase Bank especially after 1980. But one thing that I have to keep in mind is that since my Bank is rather new and or with very recent history; it might not be able to contend or be able to acquire as easily as it was for JP Morgan in my previous life.

So, I have to let my name be resounded in US media/ newspaper in order to increase weight/value of anything associated with me. It will not only raise the valuation of my companies but also will make my path for bank acquisition smoother as Genius tag on me will be able to partially compensate for our lack in various areas of banking industry henceforth raise our plateform to give us more level playing field. So according to my previous plans to increase my media presence, it aligns with above mentioned goals as well.